Techniques for map correction in the parallel imaging field have been used, such as the polynomial fit procedure (Pruessmann K P W M, Scheidegger M B, Boesiger P. SENSE: Sensivity encoding for fast MRI. Magn Reson Med 1999; 42: 952-962), thin-plate splines (Miguel Angel Gonzalez Ballester Y M, Yoshimori Kassai, Yoshinori Hamamura, Hiroshi Sugimoto, Robust Estimation of Coil Sensitivities for RF Subencoding Acquisition Techniques. 2001. p 799), wavelets (F. H. Lin Y-JC, and J. W. Belliveau, L. L. Wald, Estimation of coil sensitivity map and correction of surface coil magnetic resonance images using wavelet decomposition. 2001; Brighton, UK), and Gaussian kernel smoothing (M. S. Cohen R M D, and M. M. Zeineh. Rapid and effective correction of RF inhomogeneity for high field magnetic resonance imaging. Hum Brain Mapp 2000; 10: 204-211). There are references that disclose inpainting techniques (G. Aubert L V. A variational method in image recovery. SIAM J Numer Anal 1997; 34(5): 1948-1979; Shen TFCaJ. Non-texture inpainting by curvature driven diffusions. J Visual Comm Image Rep 2001; 12(4): 436-449; Tony F Chan J S, Luminita Vese. Variational PDE models in image processing. Amer Math Soc Notice 2003; 50: 14-26; Tony F Chan J S. Mathematical model for local non-texture inpaintings. SIAM J Appl Math 2001; 62(3): 1019-1043; V. Caselles J-MM, and C. Sbert. An axiomatic approach to image interpolation. IEEE Trans Image Processing 1998; 7(3): 376-386). There are also references that teach sensitivity map correction (Pruessmann K P W M, Scheidegger M B, Boesiger P. SENSE: Sensitivity encoding for fast MRI. Magn Reson Med 1999; 42:952-962; Miguel Angel Gonzalez Ballester Y M, Yoshinori Kassai, Yoshinori Hamamura, Hiroshi Sugimoto. Robust Estimation of Coil Sensitivities for RF Subencoding Acquisition Techniques. 2001, p. 799; F.-H. Lin Y-JC, and J. W. Belliveau, L. L. Wald. Estimation of coil sensitivity map and correction of surface coil magnetic resonance images using wavelet decomposition. 2001; Brighton, UK.; M. S. Cohen RMD, and M. M. Zeineh. Rapid and effective correction of RF inhomogeneity for high field magnetic resonance imaging. Hum Brain Mapp 2000; 10:204-211, U.S. Published Application No. U.S. 2004/0000906 A1, King et al., published Jan. 1, 2004). However, these existing techniques for use in parallel imaging do not protect the local texture.
The existing methods referred to are based on the assumption that the map is sufficiently smooth (i.e., contains no sharp variations). However, the map that needs correction is typically piece-wise smooth. Hence, the map corrected by these methods is often over-smoothed and the local texture is often destroyed.
Image inpainting was originally an artistic phrase referring to an artist's restoration of a picture's missing pieces. Computer techniques could significantly reduce the time and effort required for fixing digital images, not only to fill in blank regions but also to correct for noise. Digital inpainting techniques (C. Ballester M B, V. Caselles, G. Sapiro, and J. Verdera. Filling-in by joint interpolation of vector fields and grey levels. IEEE Trans Image Process 2001; 10(8): 1200-1211; Tony F Chan J S, Luminita Vese. Variational PDE models in image processing. Amer Math Soc Notice 2003; 50:14-26; Tony F Chan J S. Mathematical model for local non-texture inpaintings. SIAM J Appl Math 2001; 62(3): 1019-1043; Shen TFCaJ. Non-texture inpainting by curvature driven diffusions. J Visual Comm Image Rep 2001; 12(4): 436-449; V. Caselles J-MM, and C. Sbert. An axiomatic approach to image interpolation. IEEE Trans Image Processing 1998; 7(3): 376-386) are finding broad applications such as, image restoration, dis-occlusion, perceptual image coding, zooming and image super-resolution, and error concealment in wireless image transmission. Due to its broad range of applications, various methods for inpainting have been developed, ranging from nonlinear filtering methods, wavelets and spectral methods, and statistical methods (especially for textures). The most recent approach to non-texture inpainting is based on the PDE method and the Calculus of Variations. According to Chan and Shen (Tony F Chan J S, Luminita Vese. Variational PDE models in image processing. Amer Math Soc Notice 2003; 50: 14-26) PDE based TV models and the Mumford-Shah model work very well for inpainting problems with a more local nature such as hole filling (holes being regions of low signal in magnetic resonance (MR) images). Hole filling is an important issue in the correction of MR sensitivity maps, which are generally derived from MR images.
In magnetic resonance imaging (MRI) systems, sensitivity maps are meant to contain the sensitivity information of radio frequency (RF) probe coils. Highly accurate knowledge of the spatial receiver sensitivity is required by both SMASH (Simultaneous Acquisition of Spatial Harmonics (Sodickson D. Spatial Encoding Using Multiple RF Coils: SMASH Imaging and Parallel MRI. I Y, editor: John Wiley & Sons Ltd; 2000.)) and SENSE (Sensitivity Encoding (Pruessmann K P W M, Scheidegger M B, Boesiger P. Coil sensitivity encoding for fast MRI. 1998; Sixth Scientific Meeting of ISMRM, Sydney, Australia. P. 579.; Pruessmann K P W M, Scheidegger M B, Boesiger P. SENSE: Sensitivity encoding for fast MRI. Magn Reson Med 1999; 42: 952-962.)). Sensitivity maps are also important to correcting the inhomogeneities of MRI surface coils. Accurate sensitivity information can only be obtained where signal is present. However, some data sets have large areas contributing little or no signal. Such dark regions (i.e. holes) are common, for example, in pulmonary MRI using Fresh Blood Imaging (FBI). In this case, sensitivity map interpolation techniques are often used to fix the holes. To deal with slightly varying tissue configurations and motion, extrapolation over a limited range can be utilized. Therefore, there is a need for technique that is capable of simultaneously interpolating and extrapolating. There are some existing techniques for this purpose, such as the polynomial fit procedure (Sodickson D. Spatial Encoding Using Multiple RF Coils: SMASH Imaging and Parallel MRI. I Y, editor: John Wiley & Sons Ltd; 2000.), thin-plate splines (Miguel Angel Gonzalez Ballester Y M, Yoshimori Kassai, Yoshinori Hamamura, Hiroshi Sugimoto, Robust Estimation of Coil Sensitivities for RF Subencoding Acquisition Techniques. 2001. p 799.), wavelets (F. H. Lin Y-JC, and J. W. Belliveau, L. L. Wald. Estimation of coil sensitivity map and correction of surface coil magnetic resonance images using wavelet decomposition. 2001; Brighton, U K.), and Gaussian kernel smoothing (M. S. Cohen RMD, and M. M. Zeineh. Rapid and effective correction of RF inhomogeneity for high field magnetic resonance imaging. Hum Brain Mapp 2000; 10: 204-2). These methods are based on the assumption that the sensitivity map is sufficiently smooth (i.e., containing no sharp variations). However, this assumption is not always true. If the sensitivity map is piecewise smooth, which is usually true for non-uniform loading, the existing methods are not sufficient. Accordingly, there is a need for an inpainting model that can apply a technique to handle interpolation and extrapolation for piecewise smooth maps simultaneously.